On Semifields of Type

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◭◭ ◮◮

◭ ◮ On semiﬁelds of type (q2n,qn,q2,q2,q), n odd page 1 / 19 ∗ go back Giuseppe Marino Olga Polverino Rocco Tromebtti

full screen Abstract 2n n 2 2 close A semifield of type (q , q , q , q , q) (with n > 1) is a finite semifield of order q2n (q a prime power) with left nucleus of order qn, right 2 quit and middle nuclei both of order q and center of order q. Semifields of type (q6, q3, q2, q2, q) have been completely classified by the authors and N. L. Johnson in [10]. In this paper we determine, up to isotopy, the form n n of any semifield of type (q2 , q , q2, q2, q) when n is an odd integer, proving n−1 that there exist 2 non isotopic potential families of semifields of this type. Also, we provide, with the aid of the computer, new examples of semifields of type (q14, q7, q2, q2, q), when q = 2.

Keywords: semiﬁeld, isotopy, linear set MSC 2000: 51A40, 51E20, 12K10

1. Introduction

A finite semifield S is a finite algebraic structure satisfying all the axioms for a skew field except (possibly) associativity. The subsets

Nl = {a ∈ S | (ab)c = a(bc), ∀b, c ∈ S} ,

Nm = {b ∈ S | (ab)c = a(bc), ∀a, c ∈ S} ,

Nr = {c ∈ S | (ab)c = a(bc), ∀a, b ∈ S} and

K = {a ∈ Nl ∩ Nm ∩ Nr | ab = ba, ∀b ∈ S}

are fields and are known, respectively, as the left nucleus, the middle nucleus, the right nucleus and the center of the semifield. A finite semifield is a vector space ∗ ACADEMIA This work was supported by the Research Project of MIUR (Italian Office for University and Re- PRESS search) “Strutture geometriche, combinatoria e loro applicazioni” and by INDAM (Istituto Nazionale di Alta Matematica “Francesco Severi”. I I G

over its nuclei and its center (for more details on semifields see e.g. [4, 9]). If S ◭◭ ◮◮ satisfies all the axioms of a semifield except possibly the existence of the identity element of the multiplication, then S is called pre-semifield. From now on the ◭ ◮ terms semifield and pre-semifield will be always used to denote a finite semifield and a finite pre-semifield. If S S is a pre-semifield, then S is an page 2 / 19 = ( , +, ◦) ( , +) elementary abelian p-group (p prime); hence, S is an Fp-vector space. Now, if S S S′ S′ ′ go back = ( , +, ◦) and = ( , +, ◦ ) are two pre-semifields whose additive groups are elementary abelian p-groups, then S and S′ are isotopic if there exist three F S S′ full screen invertible p-linear maps, f1, f2 and f3 of into such that ′ f1(x) ◦ f2(y) = f3(x ◦ y) , close for each x, y ∈ S. The dimensions of a semifield S over its nuclei and its center

quit are invariant under the isotopy relation. From any pre-semifield it is possible to construct a semifield which is isotopic to the starting pre-semifield. The sizes of the nuclei as well as the size of the center of a semifield are invariant under isotopy. Semifields coordinatize certain translation planes (called semifield planes) and two semifield planes are isomorphic if and only if the corresponding semifields are isotopic (see [1]). A semifield is isotopic to a field if and only if the corresponding semifield plane is Desarguesian.

Let b be an element of a semifield S with center K; then the map ϕb mapping x ∈ S to xb ∈ S is a linear map when S is regarded as a left vector space over Nl. The set S = {ϕb | b ∈ S} is called the spread set of linear maps of S; it is closed under the sum of linear maps, |S| = |S| and λϕb = ϕλb for any λ ∈ K, i.e. S is a K-vector subspace of the vector space V of all Nl-linear maps of S. We say that a semifield is of type (q2n, qn, q2, q2, q) (q a prime power), if it has order q2n, left nucleus of order qn, right and middle nuclei both of order q2 and center of order q. In this paper we prove that a semifield S of type (q2n, qn, q2, q2, q), n odd, is n+1 S F 2 isotopic to a semifield j =( q n , +, ◦), 2 ≤ j < n, with multiplication given by qn x ◦ y =(α1 + α2a2 + ··· + αjaj)x +(β1b1 + β2b2 + ··· + βn−jbn−j)x ,

where y = α1 + α2a2 + ··· + αjaj + β1b1 + β2b2 + ··· + βn−jbn−j (αi, βi ∈ Fq2 ) and {1, a2,...,aj, b1, b2,...,bn−j} is an Fq2 -basis of Fq2n such that n α + α a + ··· + α a q +1 1 2 2 j j 6= 1 , β b + β b + ··· + β − b − 1 1 2 2 n j n j ACADEMIA F PRESS for each αi, βi ∈ q2 , (β1,...,βn−j) 6= (0,..., 0). Moreover, by using the geo- metric properties of the spread sets of linear maps associated with such semi- ′ ′ S S ′ n+1 ﬁelds we prove that two semiﬁelds j and j with 2 ≤ j, j < n and j 6= j I I G

are not isotopic. Hence the semiﬁelds of type (q2n, qn, q2, q2, q), n odd, are par- ◭◭ ◮◮ n−1 titioned into 2 non-isotopic families.

◭ ◮ Some semifields of type S n+1 are constructed by the authors and N. Johnson 2 in [11, Section 4] generalizing the cyclic semifields [8]. Moreover, semifields page 3 / 19 of type Sn−1 belong to a family of semifields introduced in [16] and studied in [20]. The other classes seem to be likely places to search for new semifields. go back Indeed, computational results obtained using MAGMA provide some new examples of semifields of type (q14, q7, q2, q2, q), for q = 2 and j = 5. full screen

close 2. Preliminary results

quit Let S = (Fq2n , +, ◦) be a semifield two dimensional over its left nucleus Fqn , with identity element 1 and center Fq, and let S be the spread set of Fqn -linear maps of Fq2n defining the multiplication ◦, i.e. x ◦ y = ϕy(x) where ϕy is the unique element of S such that ϕy(1) = y. Since S has center Fq, we have x ◦ y = xy = y ◦ x for each x ∈ Fq and y ∈ Fq2n , and hence S contains the field of linear maps Fq = {x ∈ Fq2n 7→ αx ∈ Fq2n | α ∈ Fq}. An element z ∈ Fq2n belongs to the right nucleus of S if x ◦ (y ◦ z)=(x ◦ y) ◦ z for each x, y ∈ Fq2n , i.e. z ∈ Nr if ϕy◦z = ϕzϕy (juxtaposition stands for the composition of maps) for each y ∈ Fq2n . So the right nucleus of S defines a field Nr = {ϕz | z ∈ Nr} of linear maps contained in S isomorphic to Nr with respect to which S is a left vector space, i.e. NrS = {µϕ | µ ∈ Nr, ϕ ∈ S} = S and Nr can be characterized as the maximal field of linear maps contained in S with respect to which S is a left vector space. Similarly, the middle nucleus of S defines a field Nm = {ϕz | z ∈ Nm} of linear maps contained in S isomorphic to Nm with respect to which S is a right vector space, i.e. SNm = S and Nm can be characterized as the maximal field of linear maps contained in S with respect to which S is a right vector space.

If Ψ and Φ are invertible Fqn -linear maps of Fq2n and σ is an automorphism of Fq2n , then the set

′ S = ΨSσΦ = {ΨϕσΦ | ϕ ∈ S} (∗)

n n (where ϕσ : x 7→ aσx + bσxq for ϕ : x 7→ ax + bxq and the composition of maps is to be read from right to left) is an additive spread set of linear maps that ′ defines on Fq2n a pre-semifield S isotopic to S. Conversely, any pre-semifield ′ ′ ′ S = (Fq2n , +, ◦ ) isotopic to S is defined by a spread set S of type (∗) (see ACADEMIA e.g. [10, 11]). PRESS σ −1 N ′ Note that ΨNr Ψ is a field, isomorphic to r, with respect to which S σ −1 ′ ′ −1 σ is a left vector space, i.e. ΨNr Ψ S = S and similarly Φ NmΦ is a field, I I G

N ′ −1 σ ′ isomorphic to m, such that S Φ NmΦ = S . Hence we have the following ◭◭ ◮◮ property. ′ ′ ◭ ◮ Property 2.1. If S is a pre-semifield, with associated spread set S , isotopic to the semifield S, then the right (respectively, middle) nucleus of S is isomorphic to the page 4 / 19 maximal field K of linear maps contained in V with respect to which KS′ = S′ (respectively, S′K = S′). go back Any element ϕ ∈ V can be uniquely written as full screen qn ϕ = ϕa,b : x ∈ Fq2n 7→ ax + bx ∈ Fq2n

close qn+1 qn+1 and ϕa,b is non-invertible if and only if a = b (see [14, p. 361]). Since qn+1 qn+1 q(ϕa,b) = a − b is a quadratic form of V over Fqn , the non-invertible quit elements of V deﬁne the hyperbolic quadric

qn+1 qn+1 Q = [ϕa,b]Fqn | a − b = 0, (a, b) 6= (0, 0)

n o n of the 3-dimensional projective space P = PG(V, Fqn ) = PG(3, q ). Here the F n V symbol [ϕa,b]Fqn denotes the 1-dimensional q -vector subspace of generated by ϕa,b.

If S is the spread set of Fqn -linear maps of a pre-semiﬁeld S = (Fq2n , +, ◦) with center Fq, then S is an Fq-vector subspace of V of dimension 2n and all the non-zero elements of S are invertible maps. Therefore, the Fq-linear set S P L( ) = [ϕ]Fqn | ϕ ∈ S, ϕ 6= 0 of deﬁned by the nonzero vectors of S is disjoint from Q. Also, any semilinear map of V of type Γ: ϕ ∈ V 7→ ΨϕσΦ ∈ V , (♦)

where Ψ and Φ are invertible Fqn -linear maps of V and σ ∈ Aut(Fq2n ), induces a collineation of P preserving the reguli of Q, and conversely (see [3]). Hence: Theorem 2.2 ([3]). Two isotopic semiﬁelds two dimensional over their left nuclei deﬁne linear sets isomorphic under the action of the collineation group G ≤ PΓO+(4, qn) preserving the reguli of Q.

P S We say that a point P = [ϕ]Fqn of has weight i in L( ) if dimFq (S ∩[ϕ]Fqn ) = i (i = 0,...,n) (see e.g. [10, 19]) and we will write w(P ) = i. In a similar way, ′ P S ′ we say that a line l = [ϕ, ϕ ]Fqn of has weight i in L( ) if dimFq (S∩[ϕ, ϕ ]Fqn ) = i (i = 0,..., 2n); if i = 0, then the line l is external to L(S), while, if i = 2n, then L(S) = l. Also, the following properties concerning the weight of points ACADEMIA and lines hold true. PRESS Property 2.3 ([10, Property 3.1]). A line l of P is contained in L(S) if and only if the weight of l in L(S) is at least n + 1. I I G

Property 2.4. The weight of a point as well as the weight of a line in L(S) is ◭◭ ◮◮ S S Γ invariant under isotopy, i.e. if 1 and 2 are isotopic pre-semiﬁelds and S1 = S2, where Γ is an invertible semilinear map of V of type (♦), and P (respectively, r) is ◭ ◮ ϕ ϕ a point (respectively, a line) of P of weight i in L(S1), then P (respectively, r ) has weight in S ϕ S where is the collineation of P induced by . page 5 / 19 i L( 1) = L( 2) ϕ Γ

Proof. Two pre-semifields S and S with associated spread sets S and S , re- go back 1 2 1 2 Γ spectively, are isotopic if and only if S1 = S2, where Γ is an invertible semilinear map of V of type (♦). Now, noting that an invertible semilinear map of V pre- full screen serves the dimension of the Fq-vector subspaces, the result easily follows. close Starting from a given semifield S = (Fq2n , +, ◦) with left nucleus Fqn , two other semifields, two dimensional over their left nuclei, can be constructed: quit the transpose semifield ST and the translation dual semifield S⊥ of S. More precisely, if πS is the semifield plane coordinatized by the semifield S, then ST is ⊥ the semifield which coordinatizes the dual plane of πS; whereas S is defined by using the polarity ⊥ associated with the hyperbolic quadric Q of P as originally introduced in [15] (for more details see also [9, Chapter 85]). In [18] the following has been proved.

Theorem 2.5 ([18, Theorem 4.1]). Let S = (Fq2n , +, ◦) be a semifield with left T nucleus Fqn and let S be the associated spread set of linear maps. Then S = n ST {ϕa,bq | ϕa,b ∈ S} is a spread set defining the transpose semifield .

qn+1 qn+1 The polar form associated with the quadratic form q(ϕa,b) = a − b qn ′ ′qn qn ′ ′qn of V is σ(ϕa,b, ϕa′,b′ ) = a a + a a − b b − b b. Hence hϕa,b; ϕa′,b′ i = Trqn/q(σ(ϕa,b, ϕa′,b′ )), where Trqn/q is the trace function of Fqn over Fq, is a non-degenerate bilinear form of V over Fq and

⊥ S = {ϕa′,b′ |hϕa,b; ϕa′,b′ i = 0 ∀ ϕa,b ∈ S}

is the orthogonal complement of S with respect to h· ; ·i. By [18, Section 3], ⊥ S is a spread set of Fqn -linear maps of Fq2n deﬁning a pre-semiﬁeld isotopic to the translation dual S⊥ of S.

We end this section by recalling the following property.

Property 2.6. If l is a line of P of weight j in L(S), then l⊥ has weight j in L(S⊥) as well.

ACADEMIA Proof. The statement follows from Equality (3) of [19, Section 2.1] in the case PRESS r = 4, t = 2n and i = 1. I I G

3. The main Theorem ◭◭ ◮◮

◭ ◮ We start by stating the following lemma.

Lemma 3.1. Let F be a set of Fqn -linear maps of Fq2n , n odd, such that page 6 / 19 (i) F is a ﬁeld of order q2 with respect to the sum and the composition of linear go back maps;

(ii) F contains the ﬁeld of linear maps Fq = {x ∈ Fq2n 7→ αx ∈ Fq2n | α ∈ Fq} . full screen Then there exists an invertible Fqn -linear map Ψ of Fq2n such that

close −1 Ψ F Ψ = Fq2 = {x ∈ Fq2n 7→ ηx ∈ Fq2n | η ∈ Fq2 } .

quit Proof. A slight generalization of [11, Lemma 2.1].

In light of this result, we are able to prove the main Theorem of the paper.

Theorem 3.2. Let S be a semiﬁeld of type (q2n, qn, q2, q2, q), n odd. Then S is n+1 S F 2 isotopic to a semiﬁeld j = ( q n , +, ◦), 2 ≤ j < n, with multiplication given by

qn x ◦ y =(α1 + α2a2 + ··· + αjaj)x +(β1b1 + β2b2 + ··· + βn−jbn−j)x , (1)

where y = α1 + α2a2 + ··· + αjaj + β1b1 + β2b2 + ··· + βn−jbn−j (αi, βi ∈ Fq2 ) and {1, a2,...,aj, b1, b2,...,bn−j} is an Fq2 -basis of Fq2n such that

n α + α a + ··· + α a q +1 1 2 2 j j 6= 1 , (2) β b + β b + ··· + β − b − 1 1 2 2 n j n j

for each αi, βi ∈ Fq2 , (β1,...,βn−j) 6= (0,..., 0).

Conversely, if {1, a2,...,aj, b1, b2,...,bn−j} is an Fq2 -basis of Fq2n (n odd and n+1 S F 2 2 ≤ j < n) satisfying (2), then the algebraic structure j = ( q n , +, ◦) where ◦ is deﬁned as in (1), is a semiﬁeld of type (q′2t, q′t, q′2, q′2, q′), where q′ = qs, n = st and s | gcd(n, j) .

Proof. Let S be a semifield of type (q2n, qn, q2, q2, q), n odd. Since S has order q2n, left nucleus of size qn and center of size q, we may assume that S =(Fq2n , +, ◦), Nl = Fqn and hence K = Fq. Then the spread set S associated with S contains the field of linear maps Fq and the right and the middle nuclei of S determine two subsets Nr and Nm of V which are fields of linear maps of ACADEMIA order q2 containing the field F and such that N S = S and SN = S. By the PRESS q r m previous lemma there exist two invertible Fqn -linear maps of Fq2n , Φ and Ψ, I I G

−1 −1 ′ ′ such that Φ NrΦ = Ψ NmΨ = Fq2 . So the pre-semiﬁeld S = (Fq2n , +, ◦ ) ◭◭ ◮◮ ′ −1 deﬁned by the set of Fqn -linear maps S = Φ SΨ is isotopic to S, and since

◭ ◮ ′ −1 −1 ′ Fq2 S = Φ NrSΨ = Φ SΨ = S

page 7 / 19 and ′ −1 −1 ′ S Fq2 = Φ SNmΨ = Φ SΨ = S , go back the pre-semiﬁeld S′ is a left and a right vector space over the ﬁeld of linear ¯ F full screen maps Fq2 . Hence we have that, if β : x 7→ βx, with β ∈ q2 , then ¯ ¯ ′ close βϕ, ϕβ ∈ S ,

′ qn for all ϕ ∈ S . Therefore, if ϕ = ϕa,b : x 7→ ax + bx , then quit n βϕ¯ − ϕβ¯: x 7→ b(β − βq)xq

′ qn ′ is an element of S for each β ∈ Fq2 . This implies that x 7→ bx belongs to S ′ as well as the map x 7→ ax. Then we can write S = S1 ⊕ S2, where S1 is an Fq2 -vector subspace of

D = {x 7→ ax | a ∈ Fq2n } = {aI | a ∈ Fq2n } (with I : x ∈ Fq2n 7→ x ∈ Fq2n )

and S2 is an Fq2 -vector subspace of

′ qn D = {x 7→ bx | b ∈ Fq2n } = {bJ | b ∈ Fq2n } qn (with J : x ∈ Fq2n 7→ x ∈ Fq2n ) .

′ Since S is an n-dimensional Fq2 -vector subspace of V, then we have that if

dimF 2 S1 = j, then dimF 2 S2 = n − j. So, if S1 = [a1I,...,ajI]F 2 and S2 = q q ′ q − F , we can write the spread set in the following way: [b1J,...,bn jJ] q2 S

′ S = x 7→ (α1a1 + ··· + αjaj) x qn F +(β1b1 + ··· + βn−jbn−j) x | αi, βi ∈ q2 . (3)

The linear map Γ: ϕa,b ∈ V 7→ ϕa,bJ = ϕb,a ∈ V defines a pre-semifield isotopic to S′. By these arguments we may assume that, up to isotopy, S′ is of type (3) n+1 ′ F 2 with j ≥ 2 . Note that if j = n, then S = D = {aI | a ∈ q n } and hence ′ n+1 F 2 the spread set S defines the field q n ; so in our hypothesis 2 ≤ j < n. −1 ′ −1 ′ Also, the spread set a1 S = {a1 ϕ | ϕ ∈ S } contains the identity map I and ACADEMIA defines a semifield isotopic to S′. Hence we may suppose, up to isotopy, that PRESS ′ a1 = 1. Finally, since all the non-zero maps of S are invertible we easily get condition (2). I I G

Now, suppose that {1, a2,...,aj, b1, b2,...,bn−j} is an Fq2 -basis of Fq2n ◭◭ ◮◮ n+1 ( 2 ≤ j < n) satisfying (2). Then the map

◭ ◮ ϕα1,...,αj ,β1,...,βn−j : x 7→ (α1 + α2a2 + ··· + αjaj) x qn − − page 8 / 19 +(β1b1 + ··· + βn jbn j) x

is non-singular for any αi, βi ∈ Fq2 not all zero. So go back F 2 Sj = {ϕα1,...,αj ,β1,...,βn−j | αi, βi ∈ q } full screen is an additive spread set of Fqn -linear maps and it defines a semifield Sj = (Fq2n , +, ◦), where ◦ is defined as in (1), with left nucleus Fqn . Recall that by close Property 2.1, the right nucleus of Sj is isomorphic to the maximal field K of linear maps contained in V such that KSj = Sj. Hence, since Fq2 Sj = Sj, quit qn we have Fq2 ⊆ K. Now, let ϕ : x 7→ Ax + Bx be any element of K. Then F 2 ϕϕα1,...,αj ,0,...,0 ∈ Sj for any αi ∈ q , and, since j > n − j, this implies B = 0. ¯ Hence K ⊂ D and this implies that K = Fq2s = {A : x → Ax | A ∈ Fq2s } for S F 2s ¯ some s | n, so the right nucleus of j is q . Also, since Aϕα1,...,αj ,β1,...,βn−j = for each F 2s , we get that F and Aϕα0,...,αj ,β1,...,βn−j ∈ Sj A ∈ q [1, a2,...,aj] q2 − F are F 2s -subspaces of F 2n and hence and . [b1, b2,...,bn j] q2 q q s | j s | (n − j) From these arguments we easily get that K also is the maximal field of linear maps contained in V such that SjK = Sj, i.e. the middle nucleus of Sj is Fq2s ′ s as well. Then the semifield Sj has center Fq′ where q = q and hence, if n = st, ′2t ′t ′2 ′2 ′ Sj is a semifield of type (q , q , q , q , q ).

S n+1 We will denote by j ( 2 ≤ j < n) a semiﬁeld whose multiplication is deﬁned as in (1) and by Sj the associated spread set of Fqn -linear maps. Remark 3.3. Note that the spread set

Sj = {x 7→ (α1 + α2a2 + ··· + αjaj) x qn +(β1b1 + ··· + βn−jbn−j) x | αi, βi ∈ Fq2 } can be written in the following way:

Sj = {x 7→ (α1 + α2a2 + ··· + αjaj) x ′ ′ qn F + b(β1 + β2b2 + ··· + βn−jbn−j) x | αi, βi ∈ q2 } , ′ where b = b and b = bk for k = 2,...,n − j. Hence condition (2) in Theo- 1 k b1 rem 3.2 can be rewritten as

qn+1 qn+1 α1 + ··· + αjaj ACADEMIA b ∈/ ′ αi, βi ∈ Fq2 , PRESS ( β1 + ··· + βn−jbn−j !

(β1,...,βn−j) 6= (0,..., 0) . (4)

) I I G

′ In what follows, we will show that two semiﬁelds Sj and Sj′ , with j 6= j ◭◭ ◮◮ ′ n+1 (j, j ≥ 2 ) are not isotopic. To this aim we start by proving the following lemma. ◭ ◮ S n+1 S Lemma 3.4. The linear set L( j) (j ≥ 2 ) associated with the semiﬁeld j page 9 / 19 contains a unique line of P and such a line has weight j in L(Sj).

⊥ go back Proof. Let r and r (where ⊥ is the polarity induced by the quadric Q) be the lines of P deﬁned by the 2-dimensional Fqn -subspaces D = {aI | a ∈ Fq2n } and ′ full screen D = {bJ | b ∈ Fq2n } of V, respectively. Since

close Sj = {x 7→ (α1 + α2a2 + ··· + αjaj) x qn +(β1b1 + ··· + βn−jbn−j) x | αi, βi ∈ Fq2 } , quit ⊥ r and r have weight 2j and 2(n − j) in L(Sj), respectively. More precisely, ′ F and F , while S1 = D ∩ Sj = [I, a2I,...,ajI] q2 dim q S1 = 2j S2 = D ∩ Sj = − F and F . [b1J, b2J,...,bn jJ] q2 dim q S2 = 2(n − j)

We ﬁrst prove that any point of r has weight at most j in L(Sj). Indeed, ∗ F F S let P = [vI] qn (with v ∈ q2n ) be a point of r with weight i in L( j), i.e., F dimFq ([vI]Fqn ∩ S1) = i and hence Sv = S1 ∩ [vI]Fqn is an q-vector subspace F of D of dimension i over q. So we can write Sv = [λ1vI,...,λivI]Fq ⊆ [vI]Fqn , where λ1,...,λi ∈ Fqn are linearly independent over Fq. Since Sv ⊆ S1 and S1 is an F 2 -subspace, we get F F ; moreover, since q [Sv] q2 = [λ1vI,...,λivI] q2 ⊆ S1 n is odd, λ1,...,λi are linearly independent over Fq2 , as well. This implies that

F F F 2i = dim q [Sv] q2 ≤ dim q S1 = 2j , ⊥ i.e. i ≤ j. In a similar way we get that the weight of any point of r in L(Sj) is at most n − j.

Now, as the line r has weight 2j in L(Sj) and 2j ≥ n + 1, by Property 2.3, r is contained in L(Sj). By way of contradiction suppose that there exists a line ℓ = PG(W, Fqn ) of P different from r contained in L(Sj). Then, by Property 2.3, S ℓ has weight t = dimFq (Sj ∩ W ) ≥ n + 1 in L( j). If ℓ ∩ r = ∅,

2n = dimFq Sj ≥ dimFq [Sj ∩ W, Sj ∩ D]Fq = t + 2j ≥ 2n + 2 ,

a contradiction. Hence ℓ∩r 6= ∅. Let π = PG(U, Fqn ) be the plane of P containing ℓ = PG(W, Fqn ) and r = PG(D, Fqn ) and let P = l ∩ r. From the previous ⊥ arguments it follows that w(P ) = dimFq (Sj ∩ D ∩ W ) ≤ j and if Q = r ∩ π, ′ then w(Q) = dimFq (Sj ∩ D ∩ U) ≤ n − j. Since ACADEMIA PRESS dimFq (Sj ∩ U) ≥ dimFq [Sj ∩ D,Sj ∩ W ]Fq

= dimFq (Sj ∩ D) + dimFq (Sj ∩ W ) − w(P ) = 2j + t − w(P ) , I I G

we have ◭◭ ◮◮ ′ 2n = dimFq Sj = dimFq [Sj ∩ U, Sj ∩ D ]Fq ◭ ◮ ′ = dimFq (Sj ∩ U) + dimFq (Sj ∩ D ) − w(Q)

page 10 / 19 ≥ 2j + t − w(P )+2(n − j) − w(Q) ≥ 2j + n + 1 − j + 2(n − j) − n + j = 2n + 1 , go back a contradiction. full screen We are now able to prove the following theorem. close ′ ′ S S ′ n+1 Theorem 3.5. Two semiﬁelds j and j , with j 6= j (j, j ≥ 2 ) are not isotopic. quit ′ Proof. By way of contradiction, suppose that Sj and Sj′ , with j 6= j (where ′ n+1 j, j ≥ 2 ) are isotopic. Then there exists an invertible semilinear map Γ of Γ ′ P type (♦) such that Sj = Sj . If ϕ is the collineation of induced by Γ, then ϕ L(Sj) = L(Sj′ ) and by Lemma 3.4 ϕ ﬁxes the line r. This means that, by ϕ ϕ Property 2.4, the line r = r has weight j in L(Sj) = L(Sj′ ), a contradiction.

By Theorems 3.2 and 3.5 the semiﬁelds of type (q2n, qn, q2, q2, q), n odd, are n−1 partitioned into not isotopic (potential) families: G n+1 (q, n), G n+3 (q, n),..., 2 2 2 Gn−1(q, n), according with the form of their multiplication as explained in The- orem 3.2.

n+1 Theorem 3.6. The families Gj(q, n), 2 ≤ j ≤ n − 1, are closed under the transpose and the translation dual operations.

n+1 S F 2 Proof. Let j = ( q n , +, ◦) be a semiﬁeld belonging to Gj(q, n) ( 2 ≤ j ≤ n − 1). Since the transpose operation leaves invariant the order of the left nucleus and interchanges the order of the right and middle nuclei [17], while the translation dual operation leaves invariant the order of the nuclei [18, The- ST S⊥ 2n n 2 2 orem 5.3], then j and j are semiﬁelds of type (q , q , q , q , q) as well. ST Also, by Theorem 3.2 and by Theorem 2.5 we get that j belongs to the family S⊥ Gj(q, n). Similarly, by Property 2.6 and Lemma 3.4, L( j ) contains a unique P S⊥ S⊥ line of and such a line has weight j in L( j ). Thus j belongs to the family Gj(q, n) as well.

In the case n = 3, it turns out that we have a unique family of semifields of ACADEMIA 2n n 2 2 PRESS type (q , q , q , q , q), namely, G2(q, 3). There exist examples of such semifields for any value of q; moreover semifields belonging to G2(q, 3) were completely classified in [10]. I I G

By Theorem 3.2 and by Remark 3.3, there exists a semifield of type Sj = ◭◭ ◮◮ F F ′ ′ ( q2n , +, ◦) (n odd) if there exists an q2 -basis {1, a2,...,aj, b, bb2,...,bbn−j} of F 2n satisfying condition (4). In [11], it has been proven that if u is an ◭ ◮ q element of Fqn not belonging to any proper subfield of Fqn and b is an element qn+1 2 of F 2n such that F with either page 11 / 19 q b = A + Bu + Cu (A, B, C ∈ q) C = 0 2 and B 6= 0 or C 6= 0 and the polynomial f(x) = A + Bx + Cx ∈ Fq[x] n−1 n−3 F 2 2 2 2 go back having two distinct roots in q, then {1, u, u ,...,u , b, bu, bu ,...,bu } is an Fq2 -basis of Fq2n satisfying condition (4) and the corresponding semifield full screen S n+1 has center Fq. Hence we have the following theorem. 2

Theorem 3.7. The family G n+1 (q, n) is not empty for any odd n and for any value close 2 of q. quit The examples exhibited above (JMPT semifields of List (L)) are obtained in [11] generalizing the cyclic semifields (JJ semifields of List (L)), and they are either cyclic semifields or isotopic to cyclic semifields. However, in the same paper, by using the computer algebra software MAGMA, two new examples of 10 10 semifields of type S n+1 for n = 5 of orders 2 and 4 with centers F2 and F4, 2 respectively, have been exhibited (JMPT(45,165) semifields of (L)). Such examples are neither cyclic nor isotopic to cyclic semifields. In [16] a potential family of semifields of type (q2n, qn, q2, q2, q), n odd, has been introduced and in [20, Section 2] it has been proved that such semifields are of type Sn−1. Also in [20], examples of semifields belonging to Gn−1(q, n) for n = 5 and q = 2 are exhibited (MT semifields of List (L)). If n ≥ 7, Theorem 3.2 provides other potential families of new semifields of type (q2n, qn, q2, q2, q), n odd. Indeed, we will prove that any possible semifield belonging to G n+3 , G n+5 ,..., Gn−2 would be new. More precisely, we show any 2 2 such semifield would not be isotopic to any known semifield nor isotopic to any derivative of a known semifield. Here, a derivative of a semifield S is, up to isotopy, a semifield obtained from S either by a Knuth operation (see [13]) or by the translation dual operation. Below we list the known examples of semifields (the classes are not neces- sarily disjoint, see C and D). This list comes from [9, Chapter 37].

ACADEMIA PRESS I I G

(L) LIST OF KNOWN SEMIFIELDS

◭◭ ◮◮ B Knuth binary commutative semiﬁelds

◭ ◮ F Flock semiﬁelds and their 5th cousins: F1 Kantor-Knuth

page 12 / 19 F2 Cohen-Ganley, 5th cousin: Payne-Thas.

F3 Penttila-Williams symplectic semifield order 35, 5th cousin, go back Bader, Lunardon, Pinneri flock semifield full screen C Commutative semifields/symplectic semifields. C1 Kantor-Williams Desarguesian Scions (symplectic), Kantor- close Williams commutative semifields

2 quit C Ganley commutative semifields and symplectic cousins C3 Coulter-Matthews commutative semifields and symplectic cousins D Generalized Dickson/Knuth/Hughes-Kleinfeld semifields S Sandler semifields JJ Jha-Johnson cyclic semifields (gen. Sandler, also of type S(ω, m, n)) JMPT Johnson-Marino-Polverino-Trombetti semifields (generalizes Jha- Johnson type S(ω, 2, n)-semifields) JMPT(45, 165) Johnson-Marino-Polverino-Trombetti non-cyclic semifields of order 45 and order 165 T Generalized twisted fields JH Johnson-Huang 8 semifields of order 82 CF Cordero-Figueroa semifield of order 36

Recently, in [5], [6], [7] and [20] the following semifields have been constructed. EMPT of order q2n, n odd Ebert-Marino-Polverino-Trombetti semifields of type (q2n, qn, q, q, q) for any odd integer n > 2 and any prime power q EMPT of order q2n, n even Ebert-Marino-Polverino-Trombetti semifields of type (q2n, qn, q2, q2, q) for any even integer n > 2 and any odd prime power q EMPT of order q6 Ebert-Marino-Polverino-Trombetti semifields of type ACADEMIA 6 3 2 6 3 2 PRESS (q , q , q , q, q) and semifields of type (q , q , q, q , q) for any odd prime power q MT Marino-Trombetti semifield of order 210 I I G

We notice that the last example belongs to the family Gn−1(q, n) for q = 2 ◭◭ ◮◮ and n = 5.

◭ ◮ Theorem 3.8. Semifields belonging to G n+3 (q, n), G n+5 (q, n), ..., Gn−2(q, n) 2 2 (with n odd ≥ 5) are new, if they exist. page 13 / 19 Proof. First we prove that no semifield of the previous list nor any derivative go back of such a semifield is of type (q2n, qn, q2, q2, q), n ≥ 5 odd, apart from the JJ, JMPT, JMPT(45, 165) and MT semifields. full screen Recall that the Knuth operations permute the nuclei of a given semifield with certain rules as shown in [17] and that the translation dual operation leaves the close sizes of the nuclei invariant [18]. A symplectic semifield and hence a flock semifield (which is the translation dual of a symplectic semifield two-dimensional quit over its left nucleus) has right and middle nuclei both isomorphic to the center [12, 17]; whereas a semifield isotopic to a commutative semifield has left and right nuclei both isomorphic to the center. Hence no semifield of type B, F, C listed above nor any of their derivatives is of type (q2n, qn, q2, q2, q). Since a Knuth semifield D of type (17), (18) or (19) (see [4, p. 241]) is 2-dimensional over at least two of its nuclei and since a Knuth semifield of type (20) (see [4, p. 242]) has the three nuclei equal to the center, no Knuth semifield of type (17), (18), (19) or (20) nor any of their derivatives is of type (q2n, qn, q2, q2, q). Straightforward computations show that the Knuth operations map a generalized Dickson semifield (see [4, p. 241, multiplication (15)]) to a generalized Dickson semifield. So, to prove that none of the derivatives of a generalized Dickson semifield is of type (q2n, qn, q2, q2, q) (n odd), it suffices to show that a generalized Dickson semifield which is two dimensional over its left nucleus, is not of type (q2n, qn, q2, q2, q) (n odd). To this aim, observe that a general- 2 ized Dickson semifield of order q is 2-dimensional over its left nucleus Fq when α = id. If σ = β and σ = β−1 then multiplication (15) coincides with the multiplication of a Knuth semifield of type (18). Hence, let either σ 6= β or σ 6= β−1. In this case, we easily get that the nuclei of the generalized Dickson semifield −1 are as follows: Nr = Fix(σβ) 6 Nl = Fq, Nm = Fix(βσ ) 6 Nl = Fq and ′2n ′n ′2 ′2 ′ K = Fq ∩Fix(σ)∩Fix(β). Such a semifield can not be of type (q , q , q , q , q ) for any n odd. 2 Next, a Sandler semifield has order qm , with left nucleus and center of order q (see [4, p. 243] and [21, Thm. 1]); hence, again by comparing the nuclei, one can see that Sandler semifields and their derivatives are not of type ACADEMIA (q2n, qn, q2, q2, q) (n odd). PRESS Also, the multiplication of a generalized twisted field of order q depends on two automorphisms of Fq, say S and T with S 6= id, T 6= id and S 6= T I I G

−1 and |Nl| = | Fix T |, |Nr| = | Fix S| and |Nm| = | Fix ST | (see [1, Lem- ◭◭ ◮◮ ma 1]). If either a generalized twisted field or any of its derivatives were of type (q2n, qn, q2, q2, q) (n odd), it would have order s2n (n odd), two of its ◭ ◮ nuclei would have order s2 and the third nucleus would have order sn; and this is not possible. Finally, JH and CF semifields are not of type 2n n 2 2 page 14 / 19 (q , q , q , q , q) with (n ≥ 5 odd) because of their orders. 5 5 go back Now, recall that JJ, JMPT and JMPT(4 , 16 ) semifields belong to G n+1 (q, n) 2 and that the MT semifield belongs to Gn−1(q, n). So, by these arguments and by full screen Theorems 3.2 and 3.6, the assertion now follows.

close 4. The question of isotopisms quit S S′ n+1 From Lemma 3.4, it follows that if j and j ( 2 ≤ j ≤ n − 1) are two isotopic semiﬁelds of the family Gj(q, n), then there exists an element ϕ of the group S ϕ S′ ϕ P G such that L( j) = L( j) and r = r, where r is the line of deﬁned by D = {aI | a ∈ Fq2n }. Hence, as in [5, Prop. 5.2 and Prop. 5.4], we get the following result.

Theorem 4.1. The spread sets

Sj = {x 7→ (α1 + α2a2 + ··· + αjaj) x qn + b(β1 + β2b2 + ··· + βn−jbn−j) x | αi, βi ∈ Fq2 }

and

′ ′ ′ Sj = {x 7→ (α1 + α2a2 + ··· + αjaj) x ′ ′ ′ qn F + b (β1 + β2b2 + ··· + βn−jbn−j) x | αi, βi ∈ q2 } F∗ F∗ deﬁne isotopic semiﬁelds if and only if there exist λ ∈ q2n , M ∈ q2n and σ ∈ Aut(Fq2n ) such that

σ σ ′ ′ F F and λ[1, a2 ,...,aj ] q2 = [1, a2,...,aj] q2 σ qn−1 σ σ ′ ′ ′ F F λb M [1, b2 ,...,bn−j] q2 = b [1, b2,...,bn−j] q2 .

5. Computational results

ACADEMIA PRESS In this final section we prove that there exist examples of semifields of type 2n n 2 2 (q , q , q , q , q), n odd, not belonging to G n+1 and Gn−1; indeed we will pro- 2 vide some examples of semifields belonging to Gn−2(q, n) for q = 2 and n = 7. I I G

By Theorem 3.2 and Remark 3.3, if n = 7 a semiﬁeld belonging to the family ◭◭ ◮◮ G5(q, 7) is of type S5 =(Fq14 , +, ◦) with multiplication given by

◭ ◮ q7 x ◦ y =(α1 + α2a2 + α3a3 + α4a4 + α5a5)x + b(β1 + β2B)x , (5)

page 15 / 19 where ai,b,B ∈ Fq14 such that

go back {1, a2,...,a5, b, bB} is an Fq2 -basis of Fq14 , (6) q7+1 ( N(b) = b ∈/ P (1, a2 ...a5,B) , full screen where close q7+1 α1 + α2a2 + ··· + α5a5 F quit P (1, a2,...,a5,B) = αi, βi ∈ q2 , ( β1 + β2B !

(β1, β2) 6= (0, 0) . )

′ α+βB ∗ F 2 F Let B = γ+δB with α,β,γ,δ ∈ q such that αδ − βγ 6= 0. If λ ∈ q2 , σ = id and b′ = b(γ + δB), we get

F F λ[1, a2, a3,...,a5] q2 = [1, a2, a3,...,a5] q2 and α + βB ′ ′ F λb[1,B] = λb(γ + δB) 1, = b [1,B ] q2 . γ + δB F q2 S′ Hence it follows from Theorem 4.1 that a semifield 5 defined by the basis ′ ′ ′ {1, a2,...,a5, b , b B }, if it exists, would be isotopic to the semifield S5 defined by the basis {1, a2,...,a5, b, bB}.

Now let q = 2 and ω be a primitive element of F27 with minimal polynomial 7 x + x + 1 ∈ F2[x] and let z be a primitive element of F214 with minimal poly- 14 7 5 3 nomial x + x + x + x + 1 ∈ F2[x]. We look for elements B ∈ F214 \ F22 for F∗ 2 3 4 which there exists b ∈ 214 such that N(b) ∈/ P (1,ω,ω ,ω ,ω ,B). If B is such F∗ 2 3 4 S an element and b ∈ 214 with N(b) ∈/ P (1,ω,ω ,ω ,ω ,B), we denote by ω,b,B ′ α+βB the corresponding semifield. By the previous arguments, for any B = γ+δB , ′ with α,β,γ,δ ∈ F22 and αδ −βγ 6= 0, the semifield Sω,b′,B′ , with b = b(γ +δB), is isotopic to Sω,b,B. ′ ′ Given two elements B,B ∈ F214 \ F22 , we say that B and B are F22 -equiva- ′ α+βB F 2 lent if there exist α,β,γ,δ ∈ 2 with αδ − βγ 6= 0 such that B = γ+δB . Note ACADEMIA that such a relation is an equivalence relation. PRESS In light of these remarks, MAGMA computations [2] show that the elements F F S F∗ B ∈ 214 \ 22 producing a semifield ω,b,B for some b ∈ 214 , up to the above I I G

equivalence relation, are those listed in Table 1. For each such B there exists ◭◭ ◮◮ 2 3 4 exactly one element η ∈ F27 \P (1,ω,ω ,ω ,ω ) and such an element is listed in the second column of Table 1. So, for each element b ∈ F 14 such that N(b) = η, ◭ ◮ 2 we get a semiﬁeld of type Sω,b,B.

page 16 / 19 B N(b) 1647 21 go back B1 = z η1 = ω 106 40 B2 = z η2 = ω full screen 122 50 B3 = z η3 = ω B = z441 η = ω19 close 4 4 Table 1 quit

Also, if Bi 6= Bj (i, j ∈ {1, 2, 3, 4}), then Bi and Bj are not F22 -equivalent. Note ′ F∗ ′ F∗ that, if b, b ∈ 214 such that N(b) = N(b ), then there exists M ∈ 214 with 7 ′ 2 −1 b = M b and, hence, Sω,b,B and Sω,b′,B are isotopic (see Theorem 4.1 with λ = 1, σ = id and M = M). Then, for any i ∈ {1, 2, 3, 4}, the pair (Bi,ηi) S defines, up to isotopy, a unique semifield of type ω,bi,Bi , where N(bi) = ηi. Hence there exist at most 4 semifields of type Sω,b,B, up to isotopy.

Now suppose that the two semiﬁelds of type Sω,b,B and Sω,b′,B′ are isotopic. F∗ F∗ F Then there exist λ ∈ 214 , M ∈ 214 and σ ∈ Aut( 214 ) such that σ 2σ 3σ 4σ 2 3 4 F F (7) λ[1,ω ,ω ,ω ,ω ] q2 = [1,ω,ω ,ω ,ω ] 22 and σ 214−1 ′ ′ F F (8) λb M [1,B] q2 = b [1,B ] 22 . If σ : x 7→ x2, condition (7) becomes 2 4 6 8 2 3 4 F F λ[1,ω ,ω ,ω ,ω ] 22 = [1,ω,ω ,ω ,ω ] 22 , and since ω7 + ω +1=0 it is equivalent to say that 2 4 6 2 3 4 F F λ[1,ω,ω ,ω ,ω ] 22 = [1,ω,ω ,ω ,ω ] 22 , and this easily implies that λ = 0, a contradiction. In a similar way, we get a contradiction for any σ ∈ Aut(F214 ) with σ 6= id. On the other hand, if σ = id, straightforward computations show that if conditions (7) and (8) are satisﬁed then

∗ ′ α + βB ′ 214−1 λ ∈ F 2 , B = , b = bM (γ + δB) , ACADEMIA 2 γ + δB PRESS ′ where α,β,γ,δ ∈ F22 and αδ − βγ 6= 0, i.e. B and B are F22 -equivalent. So we get the following result. I I G

Theorem 5.1. The semiﬁelds Sω,b,B = (F214 , +, ◦) and Sω,b′,B′ = (F214 , +, ◦) ◭◭ ◮◮ with multiplication

◭ ◮ 4 27 x ◦ y =(α1 + α2ω + ··· + α5ω ) x + b(β1 + β2B) x and 4 ′ ′ 27 page 17 / 19 x ◦ y =(α1 + α2ω + ··· + α5ω ) x + b (β1 + β2B ) x ,

go back respectively, are isotopic if and only if

′ α + βB ′ 214−1 full screen B = and b = bM (γ + δB) , γ + δB

∗ close F F 2 for some M ∈ 214 , where α,β,γ,δ ∈ 2 and αδ − βγ 6= 0 .

quit As a corollary we have: S Corollary 5.2. The semiﬁelds ω,bi,Bi , i ∈ {1, 2, 3, 4}, are pairwise non-isotopic. Also, by the previous arguments and by Theorem 3.8, we get the following result.

Theorem 5.3. In the family G5(2, 7) there exist at least four non-isotopic semi- ﬁelds, and they are new.

References

[1] A.A. Albert, Finite division algebras and ﬁnite planes, Proc. Sympos. Appl. Math. 10 (1960), 53–70.

[2] J. Cannon and C. Playoust, An introduction to MAGMA, University of Sydney Press, Sydney (1993).

[3] I. Cardinali, O. Polverino and R. Trombetti, Semiﬁeld planes of order q4 with kernel Fq2 and center Fq, European J. Combin. 27 (2006), 940–961. [4] P. Dembowski, Finite Geometries, Springer Verlag, Berlin (1968).

[5] G. Ebert, G. Marino, O. Polverino and R. Trombetti, New inﬁnite families of semiﬁelds, submitted.

[6] , On the multiplication of some semiﬁelds of order q6, Finite Fields ACADEMIA Appl., to appear. PRESS (a) [7] , Semiﬁelds in class F4 , submitted. I I G

[8] N.L. Johnson and V. Jha, An analog of the Albert-Knuth theorem on the ◭◭ ◮◮ orders of ﬁnite semiﬁelds, and a complete solution to Cofman’s subplane problem, Algebras Groups Geom. 6 no. 1 (1989), 1–35. ◭ ◮ [9] N.L. Johnson, V. Jha and M. Biliotti, Handbook of Finite Translation page 18 / 19 Planes, Pure and Applied Mathematics, Taylor Books, 2007.

go back [10] N.L. Johnson, G. Marino, O. Polverino and R. Trombetti, Semiﬁelds of 6 order q with left nucleus Fq3 and center Fq, Finite Fields Appl. 14 no. 2 full screen (2008), 456–469

close [11] , On a generalization of cyclic semiﬁelds, J. Algebraic Combin., to appear (available on line DOI 10.1007/s10801-007-0116-x).

quit [12] N.L. Johnson and O. Vega, Symplectic spreads and symplectically paired spreads, Note Mat. 26 (2006), 105–111.

[13] D.E. Knuth, Finite semiﬁelds and projective planes, J. Algebra 2 (1965), 182–217.

[14] R. Lidl and H. Niederreiter, Finite ﬁelds, Encyclopedia Math. Appl. 20, Addison-Wesley, Reading (1983) (Now distributed by Cambridge Univer- sity Press).

[15] G. Lunardon, Translation ovoids, J. Geom. 76 (2003), 200–215.

[16] , Semiﬁelds and linear sets of PG(1, qt), Quad. Mat., to appear.

[17] , Symplectic spreads and ﬁnite semiﬁelds, Des. Codes Cryptogr. 44 no. 1–3 (2007), 39–48.

[18] G. Lunardon, G. Marino, O. Polverino and R. Trombetti, Translation dual of a semiﬁeld, J. Combin. Theory Ser. A 115 (2008), no. 8, 1321– 1332.

3 [19] G. Marino, O. Polverino and R. Trombetti, Fq-linear sets of PG(3, q ) and semiﬁelds, J. Combin. Theory Ser. A 114 (2007), 769–788.

[20] G. Marino and R. Trombetti, A new semiﬁeld of order 210, submitted.

[21] R. Sandler, Autotopism groups of some ﬁnite non-associative algebras, Amer. J. Math. 84 (1962), 239–264.

ACADEMIA PRESS I I G

Giuseppe Marino ◭◭ ◮◮ DIPARTIMENTO DI MATEMATICA, SECONDA UNIVERSITA` DEGLI STUDI DI NAPOLI, VIA VIVALDI, 43, 81100 CASERTA, ITALY ◭ ◮ e-mail: [email protected]

page 19 / 19 Olga Polverino DIPARTIMENTO DI MATEMATICA, SECONDA UNIVERSITA` DEGLI STUDI DI NAPOLI, VIA VIVALDI, 43, go back 81100 CASERTA, ITALY e-mail: [email protected] full screen

Rocco Tromebtti close DIPARTIMENTO DI MATEMATICA E APPLICAZIONI,UNIVERSITA` DEGLI STUDI DI NAPOLI “FEDERICO II”, VIA CINTIA, 80126 NAPOLI, ITALY, quit e-mail: [email protected]

ACADEMIA PRESS